Reproducing kernel bounds for an advanced wavelet frame via the theta function (Q427074)

The authors consider reproducing kernels for a class of wavelets derived from multiplicatively advanced differential equations. These wavelets are based on the mother wavelet NEWLINE\[NEWLINE K(t) := \begin{cases} \displaystyle{\sum_{k\in \mathbb{Z}}} (-1)^k e^{-t q^k}/q^{k(k+1)/2}, & t\geq 0\\ 0, & t < 0, \end{cases} NEWLINE\]NEWLINE where \(q> 1\) is a parameter, and its derivatives. The reproducing kernels are stated in terms of the \(q\)-advanced sine and cosine functions \({}_q \text{Sin}\) and \({}_q \text{Cos}\), respectively. (N.B.: These functions are not related to the \(q\)-analog of the sine and cosine functions as they are not real-analytic at the origin.) Based on these results, a new family of wavelets \(\{f_k\}\) is studied, and frame properties of \(f_k\), \({}_q \text{Sin}\), and \({}_q \text{Cos}\) are established. Furthermore, it is shown that these new wavelets are Schwartz functions. Decay rates for the reproducing kernels as \(q\to\infty\) are derived, and some operator-theoretic results for the frame operator \(S\) are presented.